Closed Response Calculus and Joint Densities for the Liouville Quantum Gravity Metric
Abstract
We give a general criterion for transferring first-order responses on an underlying probability space to a closed differential calculus on the law of an observable. An integration-by-parts identity removes presentation ambiguity and yields a closable gradient, its divergence, and a closed Markov form. We apply this scheme to the subcritical Liouville quantum gravity metric throughout the range 0<γ<2. Sequential compactness of Weyl-perturbed geodesics identifies the derivative of a logarithmic distance ratio with the Sobolev Riesz representative of the difference of two normalized geodesic occupation measures. The induced form on the projective metric law is the image of a Gaussian directional form and has the energy-image-density property. Finally, fixed-target confluence and a leaf-elimination argument make the response Gram matrix positive definite for every finite forest of marked pairs. The corresponding vector of logarithmic distance ratios therefore has a Lebesgue density.
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BibTeX
@misc{cai2026closed,
title = {{Closed Response Calculus and Joint Densities for the Liouville Quantum Gravity Metric}},
author = {Chunhao Cai},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.27861},
eprint = {2609.27861},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.27861},
}
RIS
TY - UNPB
TI - Closed Response Calculus and Joint Densities for the Liouville Quantum Gravity Metric
AU - Chunhao Cai
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.27861
UR - https://arxiv.org/abs/2609.27861
AB - We give a general criterion for transferring first-order responses on an underlying probability space to a closed differential calculus on the law of an observable. An integration-by-parts identity removes presentation ambiguity and yields a closable gradient, its divergence, and a closed Markov form. We apply this scheme to the subcritical Liouville quantum gravity metric throughout the range 0<γ<2. Sequential compactness of Weyl-perturbed geodesics identifies the derivative of a logarithmic distance ratio with the Sobolev Riesz representative of the difference of two normalized geodesic occupation measures. The induced form on the projective metric law is the image of a Gaussian directional form and has the energy-image-density property. Finally, fixed-target confluence and a leaf-elimination argument make the response Gram matrix positive definite for every finite forest of marked pairs. The corresponding vector of logarithmic distance ratios therefore has a Lebesgue density.
ER -